105 of 'em. Best strategy is to get to the firehouse first and use the Axe card to give you a permanent +1 to your damage die rolls. Take advantage of the surrounding buildings and if you're in range of a specific weapon card like "Lots of Ammo" in the Sports Store or "Medkit" at the Hospital, use it and consider hunkering down to defend and let the Zombies walk to you. Remember, there is no cap on bullets but there is a 5 heart cap so don't waste a "Medkit" if you have one in play. You can use the "Medkit" card and still have it in play to use when you need it at any time.
Another good card to try and play is the "Skateboard" card so you get a permanent +2 to your movement die rolls to get yourself out of sticky situations and run for the Hospital for instance.
Will stack with "Survivalist of the Dead" and "Zombie Hunter" achievements if all 105 Zombies are killed.
Key Cards to use for Omega Man:
Alternate Food Source - Use when you need to run in a building to get hearts and bullets when your low on health.
Adrenaline Rush - +2 to combat or x2 movement roll
Fire Axe - Use in Fire Station, adds +1 to combar roll while it's in play
Hey look... A Shotgun - +1 to next 3 combat rolls
First aid Kit - Use in Hospital or Drug Store to add to in play, then can use to get +1 heart,
Grenade - USE in Army Surplus Store to add to in play, then can use to kill all zombies in building your in and you loose 1 health.
Lots of Ammo - Use in Sporting Goods Store, +3 bullets
Much Needed Rest - No movement roll, gain +2 hearts. Play before movement roll.
Skateboard - Use in SkateShop, +2 movement roll as long as it's in play.
The Keys are Still in it - Move 10 spaces, play before movement roll.